Demand modelling for emergency medical service system with multiple casualties cases: k-inflated mixture regression model

Abstract

In most of the literature on emergency medical service (EMS) system design and analysis, arrivals of EMS calls are assumed to follow Poisson process. However, it is not uncommon for real-world EMS systems to experience batch arrivals of EMS requests, where a single call involves more than one patient. Properly capturing such batch arrivals is needed to enhance the quality of analyses, thereby improving the fidelity of a resulting system design. This paper proposes a spatio-temporal demand model that incorporates batch arrivals of EMS calls. Specifically, we construct a spatio-temporal compound Poisson process which consists of a call arrival model and call size model. We build our call arrival model by combining two models available in the existing EMS demand modeling literature—artificial neural network and spatio-temporal Gaussian mixture model. For the call size model, we develop a k-inflated mixture regression model. This model reflects the characteristics of EMS call arrivals that most calls involve one patient while some calls involve multiple patients. The utility of the proposed EMS demand model is illustrated by a probabilistic ambulance location model, where we show ignoring batch arrivals leads to overestimation of ambulance availability.

Appendices

Appendix A: First and second order derivatives for negative binomial distribution and geometric distribution

Let us denote the two parameters of a negative binomial distribution by $\xi$ (number of failures) and $\delta$ (probability of success in a Bernoulli trial). Then, the first and second order derivatives of $Q_1$ and $Q_2$ for the negative binomial distribution are

$$\begin{aligned} \frac{\partial Q_1}{\partial \alpha _{ml}}= & {} \sum _{J}V_{i,t,l}[z_{i,t,m}-\pi _{i,t,m}], \\ \frac{\partial Q_2}{\partial \xi _m}= & {} \sum _{J}z_{i,t,m}[\varGamma (b_{i,t}+\xi _m-1)-\varGamma (\xi _m)+log(1-\delta _m)], \\ \frac{\partial Q_2}{\partial \delta _m}= & {} \sum _{J}z_{i,t,m}[\frac{b_{i,t}-1}{\delta _m}-\frac{\xi _m}{1-\delta _m}], \\ {\frac{\partial ^2 Q_1}{\partial \alpha _{ml}^2}}= & {} \sum _{J}V_{i,t,l}^2\pi _{im}, \\ {\frac{\partial ^2 Q_2}{\partial \xi _m^2}}= & {} \sum _{J}z_{i,t,m}[\psi (b_{i,t}+\xi _m-1)-\psi (\xi _j)], \\ {\frac{\partial ^2 Q_2}{\partial \delta _m^2}}= & {} \sum _{J}z_{i,t,m}\left[ \frac{1-b_{i,t}}{\delta _m^2}-\frac{\xi _m}{(1-\delta _m)^2}\right] , \end{aligned}$$

where $\alpha _{ml}$ indicate the coefficient for $l^{th}$ variables in $V_{i,t}$, $\varGamma$ denotes gamma function, and $\psi$ represent digamma function $\psi (x) = \frac{d}{dx}ln(\varGamma (x))$.

For geometric distribution, we use $\zeta$ to denote its parameter – success probability in Bernoulli trials. Its first and second derivatives are

$$\begin{aligned} \frac{\partial Q_1}{\partial \alpha _{ml}}= & {} \sum _{J}V_{i,t,l}[z_{i,t,m}-\pi _{i,t,m}], \\ \frac{\partial Q_2}{\partial \zeta _m}= & {} \sum _{J}z_{i,t,m}[\frac{1}{(1-\zeta _m)\zeta _m}-\frac{b_{i,t}}{1-\zeta _m}], \\ {\frac{\partial ^2 Q_1}{\partial \alpha _{ml}^2}}= & {} \sum _{J}V_{i,t,l}^2\pi _{i,t,m}, \\ {\frac{\partial ^2 Q_2}{\partial \zeta _m^2}}= & {} \sum _{J}z_{i,t,m}\left[ \frac{2\zeta _m-1}{(1-\zeta _m)^2\zeta _m^2}-\frac{b_{i,t}}{(1-\zeta _m)^2}\right] . \end{aligned}$$

Appendix B: Batch arrival hypercube queuing approximation

In this section, we briefly introduce the hypercube queuing model (Larson 1975; Jarvis 1985; Budge et al. 2009), and explain how we modify the model to incorporate batch arrivals. Consider a network of N servers that serve D demand nodes. For each demand node, servers are ordered by their preference rank by the node. Let $g_{ji}$ denote a dispatching probability of server j to demand node i when i is a $k^{th}$ preferred server. Then, $g_{ji}$ is approximated as,

$$\begin{aligned} g_{ji} \approx Q(N,\rho ,k)(1-\rho _j)\prod _{l=1}^{k-1}\rho _{a_{il}}. \end{aligned}$$

(B.1)

In Equation (B.1), $\rho$ represents the system-wise busy fraction, $\rho _j$ denotes the busy fraction of server j, and $a_{il}$ denotes the index of $l^{th}$ preferred server for demand node i. The system-wise busy fraction $\rho$ is determined as $\lambda \tau / N$, where $\lambda$ is the system-wise arrival rate and $\tau$ is the average service time. The essence of this approximation is the correction factor $Q(N,\rho ,k)$. $Q(N,\rho ,k)$ is introduced in Equation (B.1) in order to compensate for the errors due to the dependence of servers in the queuing network. Larson (1975) proposed $Q(N,\rho ,k)$ for the hypercube queue as shown below:

$$\begin{aligned} Q(N,\rho ,k) = \frac{(N-k)!P_0}{N!(1-\rho (1-P_N))(1-P_N)^{k-1}}\sum _{u=k-1}^{N-1}\frac{(N-u)(N^u\rho ^{u-k+1})}{(u-k+1)!}, \end{aligned}$$

(B.2)

where $P_0$ and $P_N$ denote the idle probability and loss probability of an M/M/N/N queue. Since the probability that u servers are busy, $P_u$, in an M/M/N/N queue is $P_u = \frac{(N\rho )^u}{u!}*P_0$, we can rewrite Equation B.2 as

$$\begin{aligned} Q(N,\rho ,k) = \frac{(N-k)!}{N!(1-\rho (1-P_N))(\rho (1-P_N))^{k-1}}\sum _{u=k-1}^{N-1}\frac{(N-u)u!}{(u-k+1)!}P_u. \end{aligned}$$

(B.3)

To account for batch arrivals, we use $P_u$ from $M^X/G/N/N$ queue (Papier and Thonemann 2008), instead of from M/M/N/N queue:

$$\begin{aligned} P_u&= \frac{e^{N\rho }\sum _{a=0}^u\frac{(N\rho )^{a}}{a!}\varPhi ^{(a)}(u)}{\sum _{u=0}^N e^{N\rho }\sum _{a=0}^u\frac{(N\rho )^{a}}{a!}\varPhi ^{(a)}(u)} \nonumber \\&= P_0 * \sum _{a=0}^u\frac{(N\rho )^{a}}{a!}\varPhi ^{(a)}(u) \end{aligned}$$

(B.4)

where $\varPhi ^{(a)}$ denotes $a^{\text {th}}$ convolution of batch size distribution $\varPhi$. That is, $\varPhi ^{(a)}(u)$ represents the probability that the sum of a batches sampled from distribution $\varPhi$ is u. Then, a generalized correction factor $Q^*(N,\rho ,k)$ is

$$\begin{aligned} Q^*(N,\rho ,k)&= \frac{(N-k)!P_0}{N!(1-\rho (1-P_N))(1-P_N)^{k-1}} \nonumber \\&\qquad \sum _{u=k-1}^{N-1}\frac{(N-u)u!}{(u-k+1)!}\sum _{a=0}^{u}\frac{N^a\rho ^{a-k+1}}{a!}\varPhi ^{(a)}(u), \end{aligned}$$

(B.5)

It is easy to show that the generalized correction factor reduces to the single arrival correction factor, Equation (B.2).

In Equation (B.5), the most challenging part is the calculation of $\varPhi ^{(a)}(u)$. Note that we use a mixture model for our call size model, i.e., $\varPhi (u)=\sum _{m\in {1,\ldots ,M}}\pi _m\phi _m(u)$, and based on the results from Sarabia et al. (2012), we derive $n^{\text {th}}$ convolution of the mixture distribution as

$$\begin{aligned} \varPhi ^{(n)}(u)=\sum _{(n_1,\ldots ,n_M)\in C_n}\frac{n!}{n_1!,\ldots ,n_M!}\pi _1^{n_1}...\pi _M^{n_M}\phi _1^{(n_1)} \otimes ...\otimes \phi _M^{(n_M)}(u), \end{aligned}$$

(B.6)

where $C_n$ is a set of combinations for n batches’ mixture membership.

In our model, we use a geometric distribution for $\phi _m$. Since the sum of random variables from geometric distribution follows a negative binomial distribution, Equation (B.6) becomes,

$$\begin{aligned} \varPhi ^{(n)}(u)&= \nonumber \\&\quad \sum _{(n_1,\ldots ,n_M)\in C_n}\frac{n!}{n_1!,\ldots ,n_M!}\pi _1^{n_1}...\pi _M^{n_M}NB(n_1,1-\zeta _1) \otimes ...\otimes NB(n_M,1-\zeta _M)(u). \end{aligned}$$

(B.7)

Furman (2007) provides an approximation to evaluate the convolution of a negative binomial distribution. (refer to Theorem 2. of Furman (2007)).